A step function changes abruptly from one constant level to another—like a stair on a staircase. Mathematically, a unit step function (a step function that rises one unit) looks like this:u(t)=0 for t<0,u(t)=1 for t≥0.
The step function is a useful input signal to measure the performance of circuits and networks. One common use is to measure the unit step response of the circuit or network under test. The step response (response to the step function) of the circuit or network can tell us information about the transfer function of the circuit or network itself. For example, Laplace transform or other mathematics can be used to predict outputs of an idealized circuit or network to an idealized unit step function. By applying a step input signal to actual circuits or networks, the circuit's performance can be measured and compared with the idealized mathematical predictions or otherwise analyzed.
One might think that switching a signal value from low to high is a simple task. But the transition time or “rise time” between low and high levels can be a challenge. In more detail, for test purposes it is often important to pay attention to the so-called “rise time” of the step function—that is, how long the signal takes to transition from the lower level (e.g., 0) to the upper level (e.g., 1). Idealized step functions have an infinitely small (zero) rise time and thus an infinitely steep (vertical on a graph) transition between the lower level and the upper level, but due to current mobility and other factors, there is always some finite delay in the real world. It is nevertheless typically desirable to make step function rise time as short as possible. Another challenge in generating a step function with a short rise is minimizing overshoot and the time the signal takes to settle to a constant level.
Commonly available function generators can generate various different signal functions including step functions with short rise times. For low-power applications, it is easy to generate the step via a set low-power voltage source, a capacitor, and a set of fast switches. Unfortunately, these kinds of signals don't necessarily possess a sufficient quantity of power to be used to test the bandwidth or other characteristics of high power circuits such as high power current sensors that may require 10 amps or more. For high-power applications, it is much more difficult because of overshoot, noise, and long settling time when a high current signal is turned on and off.
Typically, the fastest way to characterize the frequency response of a device is to expose it to an impulse or the closest approximation to the Dirac delta function. The narrower the full-width half-maximum (FWHM) of the impulse in the time domain, the wider the bandwidth it represents in the frequency domain, as the Fourier transform of the Dirac delta function is a constant across all frequencies. However, some devices or sensors are too slow to respond to this function, or if this function is produced with high-current it may not be narrow enough. In that case, it is more advantageous to realize that the Dirac delta is the derivative of the step function and expose the device or sensor to a step-function instead. Just as the Fourier transform of the derivative of the step function results in the frequency response of the impulse, the Fourier transform of the derivative of the output of the device or sensor results in the frequency response of the impulse response.
To test the bandwidth of a high power current sensor or other power circuit, it is helpful to generate a high current step-function with a fast enough rise time to generate a wide bandwidth in the frequency domain through the derivative and the subsequent Fourier transform of the resulting delta function. When a high power source is switched without appropriate signal conditioning, the result may not have a fast enough rise time, there is usually overshoot of the current, and the signal may not stabilize quickly enough to resemble a step function. The resulting signal might in fact resemble more of a ramp function with overshoot than a step function. In addition, since the step function maintains a certain current level for an amount of time specified by the design of the step function, it is also possible to obtain the steady-state response of the sensor or device under test. In summary, it enables the measurement of the frequency response and the steady-state response in one measurement.
Without a high-power step-function generator, the frequency response characterization of high-current sensors or other circuits or components may take substantially longer time to complete or might not be possible to complete, if the sensor or device is not sensitive enough to detect the lower current signal. For a basic frequency bandwidth characterization of a device, the device could be exposed to sinusoidal signals at the desired frequency(ies), typically beginning with zero until the maximum frequency of the test, and for each input signal frequency the output of the device is recorded. Finally the gain could be calculated at each frequency, resulting in the frequency response.
A faster way to perform such a measurement is to realize that the theoretically weighted sum of several sinusoids results in a step function, where the rise time depends on the frequency bandwidth (i.e., the number of frequencies) of the sum of the sinusoidal signals. The derivative with respect to time of the step function results in an impulse function. If a system is exposed to an impulse, the Fourier transform of the output is the frequency response. Alternatively, the frequency response is obtained by exposing the device to a step function comprising a large enough bandwidth of frequencies and calculating the Fourier transform of the derivative of the step-response with respect to time.